Simulations of estimating causal effects using undersmoothed HAL
Seraphina Shi
2023-05-10
Simulation 1
Data structure: \(O = (W, A, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Y - outcome, indicator of an event
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0, max=5)\)
- \(Y = \mathbf{I}[U_Y < expit(-5 + W + 2.25A -0.5WA)]\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
{r child = '2_1_Simu_results_template.Rmd'} #
Simulation 2
Data structure: \(O = (W, A, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Y - outcome, indicator of an event
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1.3^2)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2.5 - 0.5W + U_A, min=0, max=5)\)
- \(Y = \mathbf{I}[U_Y < expit(-8 + 2W + 3sin(1.5A^{1.5}) + 2WA)]\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
{r child = '2_1_Simu_results_template.Rmd'} #
Simulation 3
Data structure: \(O = (W, A, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Y - outcome, indicator of an event
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0, max=5)\)
- \(Y = \mathbf{I}[U_Y < expit(-10 - 3W + 4A + \mathbf{I}(A>2) * 5sin((0.8A)^2 - 2.6) )]\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
{r child = '2_1_Simu_results_template.Rmd'} #
Simulation 4
Data structure: \(O = (W_1, W_2, W_3, W_4, W_5, A, Y)\)
- W - baseline covariates
- A - treatment level based on W, continuous between 0 and 5
- Y - outcome, indicator of an event
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
- Structural equations F and endogenous variables:
- \(W \sim N(\mu_W, \Sigma_W)\)
- \(A = bound(0.1W_1 + 0.2W_2 + 0.5W_3 + 0.15W_4 - 0.05W_5 - 0.01W_3W_5 + U_A, min=0, max=5)\)
- \(Y = \mathbf{I}[U_Y < expit(-7 -W_1 +
2W_2 - 0.5W_4 - W_1W_3 + 4A + 0.5AW_2)]\)
- where \(\mu_W = \begin{bmatrix}0 \\0 \\5 \\1 \\1 \\\end{bmatrix}\), and \(\Sigma_W = \begin{bmatrix}1&0.8&1.35&0.1&0.03 \\0.8&1&1.2&0.15&0.03 \\1.35&1.2&2.25&0.075&0.225 \\0.1&0.15&0.075&0.25&0.1125 \\0.03&0.03&0.225&0.1125&0.09 \\\end{bmatrix}\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
{r child = '2_1_Simu_results_template_0.Rmd'} #
Simulation 5
Data structure: \(O = (W_1, W_2, W_3, W_4, W_5, A, Y)\)
- W - baseline covariates
- A - treatment level based on W, continuous between 0 and 5
- Y - outcome, indicator of an event
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
- Structural equations F and endogenous variables:
- \(W \sim N(\mu_W, \Sigma_W)\)
- \(A = bound(0.1W_1 + 0.2W_2 + 0.5W_3 + 0.15W_4 - 0.05W_5 - 0.01W_3W_5 + U_A, min=0, max=5)\)
- \(Y = \mathbf{I}[U_Y < expit(-10 -W_1 +
2W_2 - 0.5W_4 - 0.5W_1W_3 + 4A + \mathbf{I}(A>2) *5sin((0.8A)^2 -
2.6))]\)
- where \(\mu_W = \begin{bmatrix}0 \\0 \\5 \\1 \\1 \\\end{bmatrix}\), and \(\Sigma_W = \begin{bmatrix}1&0.8&1.35&0.1&0.03 \\0.8&1&1.2&0.15&0.03 \\1.35&1.2&2.25&0.075&0.225 \\0.1&0.15&0.075&0.25&0.1125 \\0.03&0.03&0.225&0.1125&0.09 \\\end{bmatrix}\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
{r child = '2_1_Simu_results_template_0.Rmd'} #
Simulation 6
Data structure: \(O = (W_1, W_2, W_3, A, Y)\)
- W - baseline covariates
- A - treatment level based on W, continuous between 0 and 1
- Y - outcome, indicator of an event
Underlying data generating process, \(P_{U,X}\)
- Structural equations F and endogenous variables:
- \(W_1 \sim Uniform(o,1)\)
- \(W_2 \sim Bernoulli(\mu=0, \sigma^2 = 2^2)\)
- \(W_3 \sim N(W_1, 0.25*exp(2W_1))\)
- \(A \sim Beta(v(W)\mu(W), v(W)[1-\mu(W)])\)
- \(Y \sim Bernoulli(Q_0(A,W))\)
- where:
- \(v(W) = exp(1 + 2W_1expit(W3))\)
- \(\mu(W) = expit(0.03 - 0.8log(1+W_2) + 0.9exp(W_1)W_2 - 0.4arctan(W_3+2)W_2W_1)\)
- \(\bar{Q}_0(A,W) = expit(-2 + 1.5A + 5A^3 - 2.5W_1 + 0.5AW_2 - log(A)W_1W_2 + 0.5A^{3/4}W_1W_3)\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in (0,1])\), the causal dose-response curve
{r child = '2_1_Simu_results_template_s6.Rmd'} #
Simulation 7
Data structure: \(O = (W, A, Y)\)
- W - baseline covariates
- A - treatment level based on W, continuous between 0 and 1
- Y - outcome, indicator of an event
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2.5 - 0.5W + U_A, min=0, max=5)\)
- \(Y = \mathbf{I}[U_Y < expit(-6 + W + 3.5A\mathbf{I}(A\geq2) - 4A\mathbf{I}(A\geq4) - 0.5WA)]\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
## W A Y
## Min. :-3.781093 Min. :0.000 Min. :0.0000
## 1st Qu.:-0.687019 1st Qu.:1.750 1st Qu.:0.0000
## Median :-0.006363 Median :2.479 Median :1.0000
## Mean :-0.011178 Mean :2.492 Mean :0.5489
## 3rd Qu.: 0.664196 3rd Qu.:3.243 3rd Qu.:1.0000
## Max. : 3.730121 Max. :5.000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 2.493 3.278 3.220 3.945 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.993 2.691 2.677 3.342 5.000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.610 2.284 2.292 2.946 5.000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.024 1.766 1.760 2.447 5.000
n = 200
CV vs Undersmoothing (Zero smoothness order with default number of knots)
## The average lambda of CV-HAL: 0.0052 (= 1 * lambda_CV )## The average lambda of globally undersmoothed HAL: 0.0004 (= 0.0681 * lambda_CV ) The average lambda of globally undersmoothed HAL: 0.0004 (= 0.0681 * lambda_CV ) The average lambda of globally undersmoothed HAL: 0.0004 (= 0.0681 * lambda_CV )## The average number of bases of CV-HAL: 10.3024 The average number of bases of CV-HAL: 10.3024## The average number of bases of globally undersmoothed HAL: 36.8775## The average fitting time for CV-HAL: 0.9538 seconds## The average fitting time for globally undersmoothed HAL: 1.4834 seconds
Undersmoothed Smoothness-adaptive HAL
## smooth_order if_n_knots_default sl_pick lambda lambda_scaler n_basis cv_risk
## 1 0.000 1 0.0000 0.0036 1.0000 6.908 0.3904
## 12 0.000 0 0.0780 0.0047 1.0000 11.148 0.2258
## 14 0.000 0 0.0780 0.0047 1.0000 11.148 0.2258
## 15 0.000 0 0.0780 0.0047 1.0000 11.148 0.2258
## 17 0.000 0 0.0780 0.0047 1.0000 11.148 0.2258
## 18 0.000 0 0.0780 0.0047 1.0000 11.148 0.2258
## 23 1.000 1 0.1880 0.0025 1.0000 8.024 0.1935
## 24 1.000 1 0.1880 0.0025 1.0000 8.024 0.1935
## 25 1.000 1 0.1880 0.0025 1.0000 8.024 0.1935
## 29 1.000 1 0.1880 0.0025 1.0000 8.024 0.1935
## 34 1.000 0 0.3722 0.0021 1.0000 12.350 0.1858
## 36 1.000 0 0.3722 0.0021 1.0000 12.350 0.1858
## 37 1.000 0 0.3722 0.0021 1.0000 12.350 0.1858
## 38 1.000 0 0.3722 0.0021 1.0000 12.350 0.1858
## 45 2.000 1 0.1520 0.0049 1.0000 9.584 0.1935
## 46 2.000 1 0.1520 0.0049 1.0000 9.584 0.1935
## 47 2.000 1 0.1520 0.0049 1.0000 9.584 0.1935
## 48 2.000 1 0.1520 0.0049 1.0000 9.584 0.1935
## 50 2.000 1 0.1520 0.0049 1.0000 9.584 0.1935
## 52 2.000 1 0.1520 0.0049 1.0000 9.584 0.1935
## 55 2.000 1 0.1526 0.0049 1.0000 9.584 0.1937
## 56 2.000 0 0.1000 0.0058 1.0000 10.846 0.1961
## 57 2.000 0 0.1000 0.0058 1.0000 10.846 0.1961
## 58 2.000 0 0.1000 0.0058 1.0000 10.846 0.1961
## 60 2.000 0 0.1000 0.0058 1.0000 10.846 0.1961
## 61 2.000 0 0.1000 0.0058 1.0000 10.846 0.1961
## 62 2.000 0 0.1000 0.0058 1.0000 10.846 0.1961
## 64 2.000 0 0.1000 0.0058 1.0000 10.846 0.1961
## 66 2.000 0 0.1008 0.0059 1.0000 10.846 0.1962
## 67 3.000 1 0.0782 0.0246 1.0000 10.008 0.2061
## 68 3.000 1 0.0782 0.0246 1.0000 10.008 0.2061
## 69 3.000 1 0.0782 0.0246 1.0000 10.008 0.2061
## 70 3.000 1 0.0782 0.0246 1.0000 10.008 0.2061
## 71 3.000 1 0.0782 0.0246 1.0000 10.008 0.2061
## 72 3.000 1 0.0782 0.0246 1.0000 10.008 0.2061
## 74 3.000 1 0.0782 0.0246 1.0000 10.008 0.2061
## 75 3.000 1 0.0783 0.0247 1.0000 10.008 0.2063
## 76 3.000 1 0.0791 0.0248 1.0000 10.008 0.2068
## 77 3.000 1 0.0822 0.0258 1.0000 10.008 0.2082
## 78 3.000 0 0.0320 0.0264 1.0000 11.026 0.2077
## 79 3.000 0 0.0320 0.0264 1.0000 11.026 0.2077
## 86 3.000 0 0.0321 0.0264 1.0000 11.026 0.2078
## 87 3.000 0 0.0324 0.0266 1.0000 11.026 0.2083
## 88 3.000 0 0.0319 0.0276 1.0000 11.026 0.2113
## 89 1.394 0.418 1.0000 0.0015 0.9014 22.494 NA
## 90 1.394 0.418 1.0000 0.0015 0.9014 22.494 NA
## 91 1.394 0.418 1.0000 0.0015 0.9014 22.494 NA
## 92 1.394 0.418 1.0000 0.0015 0.9014 22.494 NA
## 93 1.394 0.418 1.0000 0.0015 0.9014 22.494 NA
## 94 1.394 0.418 1.0000 0.0015 0.9014 22.494 NA
## 96 1.394 0.418 1.0000 0.0015 0.9014 22.494 NA
## 97 1.394 0.418 1.0000 0.0015 0.9014 22.494 NA
## 98 1.394 0.418 1.0000 0.0015 0.9026 22.494 NA
## 99 1.394 0.418 1.0000 0.0014 0.9113 22.494 NA
CV vs Undersmoothing (First smoothness order with smaller number of knots)
## The average lambda of CV-HAL: 0.0022 (= 1 * lambda_CV )## The average lambda of globally undersmoothed HAL: 0.0003 (= 0.1612 * lambda_CV )## The average number of bases of CV-HAL: 12.0630## The average number of bases of globally undersmoothed HAL: 24.0123## The average fitting time for CV-HAL: 5.3135 seconds## The average fitting time for globally undersmoothed HAL: 7.8475 seconds
grid of scalers
## [1] "proportion of simulations that cannot compute empirical SD:"## 1.2 1.1 1 0.6952 0.4833 0.336 0.2336 0.1624 0.1129 0.0785 0.0546
## 0.002 0.002 0.002 0.004 0.004 0.008 0.014 0.026 0.042 0.062 0.084
## 0.0379 0.0264 0.0183 0.0127 0.0089 0.0062 0.0043 0.003 0.0021 0.0014 0.001
## 0.106 0.162 0.212 0.302 0.388 0.472 0.544 0.604 0.698 0.754 0.808
vs GAM & Polynomial regression
n = 500
CV vs Undersmoothing (Zero smoothness order with default number of knots)
## The average lambda of CV-HAL: 0.0031 (= 1 * lambda_CV ) The average lambda of CV-HAL: 0.0031 (= 1 * lambda_CV )## The average lambda of globally undersmoothed HAL: 0.0001 (= 0.0209 * lambda_CV ) The average lambda of globally undersmoothed HAL: 0.0001 (= 0.0209 * lambda_CV )## The average number of bases of CV-HAL: 16.6580 The average number of bases of CV-HAL: 16.6580 The average number of bases of CV-HAL: 16.6580## The average number of bases of globally undersmoothed HAL: 95.8940## The average fitting time for CV-HAL: 3.5177 seconds## The average fitting time for globally undersmoothed HAL: 6.4772 seconds
Undersmoothed Smoothness-adaptive HAL
## smooth_order if_n_knots_default sl_pick lambda lambda_scaler n_basis cv_risk
## 1 0.0 1 0.0000 0.0016 1.0000 8.056 0.3784
## 12 0.0 0 0.1400 0.0026 1.0000 14.534 0.1627
## 14 0.0 0 0.1400 0.0026 1.0000 14.534 0.1627
## 18 0.0 0 0.1400 0.0026 1.0000 14.534 0.1627
## 19 0.0 0 0.1400 0.0026 1.0000 14.534 0.1627
## 20 0.0 0 0.1400 0.0026 1.0000 14.534 0.1627
## 22 0.0 0 0.1400 0.0026 1.0000 14.534 0.1627
## 23 1.0 1 0.0140 0.0008 1.0000 10.652 0.1727
## 26 1.0 1 0.0140 0.0008 1.0000 10.652 0.1727
## 27 1.0 1 0.0140 0.0008 1.0000 10.652 0.1727
## 29 1.0 1 0.0140 0.0008 1.0000 10.652 0.1727
## 31 1.0 1 0.0140 0.0008 1.0000 10.652 0.1727
## 32 1.0 1 0.0140 0.0008 1.0000 10.652 0.1727
## 34 1.0 0 0.5074 0.0006 1.0000 18.654 0.1507
## 40 1.0 0 0.5074 0.0006 1.0000 18.654 0.1507
## 42 1.0 0 0.5074 0.0006 1.0000 18.654 0.1507
## 43 1.0 0 0.5074 0.0006 1.0000 18.654 0.1507
## 45 2.0 1 0.1260 0.0012 1.0000 12.054 0.1559
## 46 2.0 1 0.1260 0.0012 1.0000 12.054 0.1559
## 47 2.0 1 0.1260 0.0012 1.0000 12.054 0.1559
## 48 2.0 1 0.1260 0.0012 1.0000 12.054 0.1559
## 49 2.0 1 0.1260 0.0012 1.0000 12.054 0.1559
## 50 2.0 1 0.1260 0.0012 1.0000 12.054 0.1559
## 51 2.0 1 0.1260 0.0012 1.0000 12.054 0.1559
## 52 2.0 1 0.1260 0.0012 1.0000 12.054 0.1559
## 53 2.0 1 0.1260 0.0012 1.0000 12.054 0.1559
## 55 2.0 1 0.1263 0.0012 1.0000 12.054 0.1559
## 56 2.0 0 0.1980 0.0012 1.0000 15.320 0.1534
## 57 2.0 0 0.1980 0.0012 1.0000 15.320 0.1534
## 58 2.0 0 0.1980 0.0012 1.0000 15.320 0.1534
## 59 2.0 0 0.1980 0.0012 1.0000 15.320 0.1534
## 60 2.0 0 0.1980 0.0012 1.0000 15.320 0.1534
## 61 2.0 0 0.1980 0.0012 1.0000 15.320 0.1534
## 62 2.0 0 0.1980 0.0012 1.0000 15.320 0.1534
## 64 2.0 0 0.1980 0.0012 1.0000 15.320 0.1534
## 65 2.0 0 0.1980 0.0012 1.0000 15.320 0.1534
## 67 3.0 1 0.0040 0.0101 1.0000 11.850 0.1679
## 76 3.0 1 0.0040 0.0101 1.0000 11.850 0.1679
## 77 3.0 1 0.0044 0.0102 1.0000 11.850 0.1674
## 78 3.0 0 0.0040 0.0096 1.0000 14.362 0.1666
## 87 3.0 0 0.0040 0.0096 1.0000 14.362 0.1665
## 88 3.0 0 0.0022 0.0096 1.0000 14.362 0.1660
## 89 1.2 0.144 1.0000 0.0006 0.8050 36.472 NA
## 90 1.2 0.144 1.0000 0.0006 0.8050 36.472 NA
## 93 1.2 0.144 1.0000 0.0006 0.8050 36.472 NA
## 97 1.2 0.144 1.0000 0.0006 0.8050 36.472 NA
## 98 1.2 0.144 1.0000 0.0006 0.8075 36.472 NA
## 99 1.2 0.144 1.0000 0.0006 0.8451 36.472 NA
CV vs Undersmoothing (First smoothness order with smaller number of knots)
## The average lambda of CV-HAL: 0.0006 (= 1 * lambda_CV )## The average lambda of globally undersmoothed HAL: 0.0001 (= 0.1644 * lambda_CV )## The average number of bases of CV-HAL: 17.7651## The average number of bases of globally undersmoothed HAL: 38.3843## The average fitting time for CV-HAL: 10.3585 seconds## The average fitting time for globally undersmoothed HAL: 20.3527 seconds
grid of scalers
## [1] "proportion of simulations that cannot compute empirical SD:"## 1.2 1.1 1 0.6952 0.4833 0.336 0.2336 0.1624 0.1129 0.0785 0.0546
## 0.002 0.002 0.002 0.004 0.000 0.012 0.014 0.030 0.068 0.136 0.196
## 0.0379 0.0264 0.0183 0.0127 0.0089 0.0062 0.0043 0.003 0.0021 0.0014 0.001
## 0.262 0.374 0.498 0.592 0.700 0.764 0.836 0.886 0.936 0.934 0.962
vs GAM & Polynomial regression
n = 1000
CV vs Undersmoothing (Zero smoothness order with default number of knots)
## The average lambda of CV-HAL: 0.0021 (= 1 * lambda_CV )## The average lambda of globally undersmoothed HAL: 0.0000 (= 0.0141 * lambda_CV ) The average lambda of globally undersmoothed HAL: 0.0000 (= 0.0141 * lambda_CV )## The average number of bases of CV-HAL: 22.4800 The average number of bases of CV-HAL: 22.4800## The average number of bases of globally undersmoothed HAL: 195.0103## The average fitting time for CV-HAL: 11.6560 seconds## The average fitting time for globally undersmoothed HAL: 21.1333 seconds
Undersmoothed Smoothness-adaptive HAL
## smooth_order if_n_knots_default sl_pick lambda lambda_scaler n_basis cv_risk
## 1 0.000 1 0.0000 0.0009 1.0000 8.630 0.3744
## 12 0.000 0 0.0200 0.0017 1.0000 17.136 0.1544
## 13 0.000 0 0.0200 0.0017 1.0000 17.136 0.1544
## 23 1.000 1 0.0000 0.0004 1.0000 12.620 0.1632
## 34 1.000 0 0.8724 0.0002 1.0000 25.174 0.1332
## 35 1.000 0 0.8724 0.0002 1.0000 25.174 0.1332
## 37 1.000 0 0.8724 0.0002 1.0000 25.174 0.1332
## 42 1.000 0 0.8724 0.0002 1.0000 25.174 0.1332
## 45 2.000 1 0.0020 0.0010 1.0000 12.658 0.1462
## 56 2.000 0 0.1040 0.0009 1.0000 16.592 0.1406
## 58 2.000 0 0.1040 0.0009 1.0000 16.592 0.1406
## 59 2.000 0 0.1040 0.0009 1.0000 16.592 0.1406
## 67 3.000 1 0.0000 0.0101 1.0000 11.888 0.1607
## 68 3.000 1 0.0000 0.0101 1.0000 11.888 0.1607
## 77 3.000 1 0.0000 0.0101 1.0000 11.888 0.1609
## 78 3.000 0 0.0000 0.0100 1.0000 14.862 0.1582
## 88 3.000 0 0.0000 0.0100 1.0000 14.862 0.1588
## 89 1.086 0.002 1.0000 0.0001 0.2675 51.896 NA
## 90 1.086 0.002 1.0000 0.0001 0.2675 51.896 NA
## 92 1.086 0.002 1.0000 0.0001 0.2675 51.896 NA
## 93 1.086 0.002 1.0000 0.0001 0.2675 51.896 NA
## 94 1.086 0.002 1.0000 0.0001 0.2675 51.896 NA
## 97 1.086 0.002 1.0000 0.0001 0.2675 51.896 NA
## 99 1.086 0.002 1.0000 0.0001 0.2688 51.896 NA
CV vs Undersmoothing (First smoothness order with smaller number of knots)
## The average lambda of CV-HAL: 0.0002 (= 1 * lambda_CV )## The average lambda of globally undersmoothed HAL: 0.0000 (= 0.1767 * lambda_CV )## The average number of bases of CV-HAL: 24.1575## The average number of bases of globally undersmoothed HAL: 52.2648 The average number of bases of globally undersmoothed HAL: 52.1975## The average fitting time for CV-HAL: 19.1449 seconds## The average fitting time for globally undersmoothed HAL: 37.1238 seconds
grid of scalers
## [1] "proportion of simulations that cannot compute empirical SD:"## 1.2 1.1 1 0.6952 0.4833 0.336 0.2336 0.1624 0.1129 0.0785 0.0546
## 0.000 0.000 0.000 0.004 0.000 0.008 0.022 0.032 0.080 0.112 0.194
## 0.0379 0.0264 0.0183 0.0127 0.0089 0.0062 0.0043 0.003 0.0021 0.0014 0.001
## 0.300 0.402 0.502 0.560 0.644 0.682 0.702 0.768 0.796 0.818 0.826
vs GAM & Polynomial regression
